Matrix elements of Lorentzian Hamiltonian constraint in loop quantum gravity

Abstract

The Hamiltonian constraint is the key element of the canonical formulation of loop quantum gravity (LQG) coding its dynamics. In Ashtekar-Barbero variables it naturally splits into the so-called Euclidean and Lorentzian parts. However, due to the high complexity of this operator, only the matrix elements of the Euclidean part have been considered so far. Here we evaluate the action of the full constraint, including the Lorentzian part. The computation requires heavy use of SUð2Þ recoupling theory and several tricky identities among n-j symbols are used to find the final result: these identities, together with the graphical calculus used to derive them, also simplify the Euclidean constraint and are of general interest in LQG computations.

Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

“…Thanks to the calculation of the matrix elements of the volume operator in certain special case, some matrix elements of Thiemann's Hamiltonian constraint operator and its generalization were derived in [21,22]. Later on, the matrix elements was re-derived in [13], and then the formula in [13] was corrected by sign factors in [14,23] using graphical method. Matter coupling is also an important issue in LQG.…”

“…Brink's graphical method was only recalled in the appendix of [13]. Then Varshalovich's method was adopted in [14,23,28]. Brink's graphical method was also taken to study the propagator of spinfoam models in [29], in which the graphical method was only used to calculate the action of the right-invariant vector field (the "grasping operator") on the intertwiners but not the action of holonomy operator.…”

“…As a powerful tool for practical calculation, graphic calculus has been introduced in LQG in a few papers (see e.g., [13,14,18,[21][22][23]28,29]). These graphical methods are based on the graphical methods developed by Yutsis in [30], Brink in [31], Varshalovich in [32], and Kauffman in [33].…”

“…Certain matrix elements of the volume operator were calculated in the framework of loop representation by using a graphical tangle-theoretic Temperley-Lieb formulation in [18]. Then they were also derived in connection representation by a rigorous but tedious algebraic method in [11,19], and their special case was re-derived using generalized Wigner-Eckart theory [20] as well as the graphical methods in [13,[21][22][23]. Although those components of the volume operator are rigorously defined, the computation of their actions on spin-network states are difficult.…”

Curvature operator for loop quantum gravity

“…It is defined employing a regularization procedure with specific rules that might be changed to bring it closer to the spinfoam formalism [17,18] (but till now spoiling the anomaly free property). However this operator is computationally extremely hard to implement [19], in particular its Lorentzian part which involves several commutators of the extrinsic curvature in order to express the Ricci scalar in terms of Holonomies and Fluxes. Few computations appeared so far [14,15] and few solutions have been found [16].…”